Regularity results for nonlocal equations and applications

Abstract

We introduce the concept of $$C^{m,\alpha }$$ -nonlocal operators, extending the notion of second order elliptic operator in divergence form with $$C^{m,\alpha }$$ -coefficients. We then derive the nonlocal analogue of the key existing results for elliptic equations in divergence form, notably the Holder continuity of the gradient of the solutions in the case of $$C^{0,\alpha }$$ -coefficients and the classical Schauder estimates for $$C^{m+1,\alpha }$$ -coefficients. We further apply the regularity results for $$C^{m,\alpha }$$ -nonlocal operators to derive optimal higher order regularity estimates of Lipschitz graphs with prescribed Nonlocal Mean Curvature. Applications to nonlocal equation on manifolds are also provided.